Brownian dynamics simulation and experimental study of colloidal particle deposition in a microchannel flow

Journal of Colloid and Interface Science 291 (2005) 28–36 www.elsevier.com/locate/jcis

Brownian dynamics simulation and experimental study of colloidal particle deposition in a microchannel flow H.N. Unni, C. Yang ∗ School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798 Received 20 November 2004; accepted 28 April 2005 Available online 17 June 2005

Abstract This paper reports an analysis of the irreversible deposition of colloidal particles from the pressure-driven flow in a microchannel within the framework of DLVO theory. A theoretical model is presented on the basis of the stochastic Langevin equation, incorporating the random Brownian motion of colloidal particles. Brownian dynamics simulation is used to compute the particle deposition in terms of the surface coverage. To validate the theoretical model, experiments are carried out using the parallel-plate flow cell technique, enabling direct videomicroscopic observation of the deposition kinetics of polystyrene latex particles in NaCl electrolytes. The theoretical predictions are compared with the experimental results, and good agreement is found.  2005 Elsevier Inc. All rights reserved. Keywords: Colloidal particle deposition; Brownian dynamics simulation; Langevin equation; Parallel-plate flow; Blocking effect

1. Introduction Deposition of colloids (e.g., particulates and macromolecules) on a solid surface is of great importance in many technological processes such as filtration, contamination control of microelectronic manufacture, control of surface fouling of microfluidic devices, and biofouling of artificial organs. In the literature, there are two theoretical approaches of modeling particle deposition on collector surfaces, the Lagrangian method and the Eulerian method [1–3]. In the Eulerian method, particle deposition on the collector surfaces is governed by the convection–diffusion equation, which also incorporates the colloidal and external forces. The Lagrangian method determines the trajectory of the particles under the effect of colloidal and external forces, and the governing equation of particle transport is the stochastic Langevin equation, including the particle Brownian motion. * Corresponding author. Fax: +65 6791 1859.

E-mail address: [email protected] (C. Yang). 0021-9797/$ – see front matter  2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2005.04.104

Particle deposition is controlled by the combined influence of colloidal and hydrodynamic interactions. In the vicinity of a channel wall, the displacement of the fluid between the particle and the wall becomes increasingly difficult because of the additional hydrodynamic drag exerted on the particle (apart from the Stokes drag of a particle in the absence of a wall). Hence, in the vicinity of a channel wall, particle motion is retarded due to the presence of the wall. Similarly, the presence of neighboring particles causes the retardation of the moving particle. These are referred to as Hydrodynamic interactions [1]. Likewise, the diffusive motion of a particle is also affected by its neighboring particles and the wall, and consequently the particle diffusivity deviates from the Stokes–Einstein diffusivity of a single particle. At closer separations from the collector surface (1–100 nm), the particle’s motion is controlled by colloidal interactions, which are constituted of by the universal van der Waals (VDW) and electrostatic double layer (EDL) interactions that form the basis of the DLVO theory of colloidal stability. Particle deposition can be evaluated by the so-called surface coverage [1]. Surface coverage is defined as the ratio

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of the area covered by the deposited particles to the surface area of the collector surface. Surface blocking is an important phenomenon that becomes prominent at later stages of deposition, when the surface is already covered with deposited particles. This phenomenon arises from the hydrodynamic and electrostatic interactions between the deposited and flowing particles, preventing further deposition. Hence surface blocking is dependent on several key parameters, such as the electrolyte concentration, particle size, and particle velocities, as these parameters determine the magnitude and range of electrostatic and hydrodynamic interactions between deposited and flowing particles. The combined effect of hydrodynamic and electrostatic interactions on surface blocking is named the “Scattering effect.” For low surface coverage (θ < 0.2), a limiting analytical expression for the surface coverage was presented by Johnson and Elimelech [4], based on Random sequential adsorption (RSA). In their work, a blocking function, which has a nonlinear dependence on the surface coverage, was introduced in contrast with the linear Langmuirian blocking function used in earlier models. Later, an extension to the RSA model using the Ballistic deposition model (BD) was considered by Senger et al. [5] for the cases in which gravity plays an important role. The validity of the RSA and BD models was discussed in detail using experimental results. For higher surface coverages, however, RSA and BD models are inadequate. To account for this, Adamczyk et al. [6] used the sequential Brownian dynamics simulation method to model the kinetics of particle deposition in an impinging jet cell. The stochastic trajectory simulation method developed by Ansell and Dickinson [7] was used to compute the particle surface coverage. The simulation results were compared with experiments using polystyrene particles, and reasonable agreement was obtained. Using Brownian dynamics simulation, Hutter [8] identified the coagulation time scales in colloidal suspensions for various solid content ratios. The dependence of the time scales on the solid content and the colloidal interaction parameters was investigated to achieve a better understanding of the coagulation mechanisms. A similar work by Villalba and Garcia-Sucre [9] addressed the stability of oil-in-water emulsions using Brownian dynamics simulation technique. Recently, the deposition of Brownian particles onto the collector surfaces of porous media was investigated by Chang et al. [10]. The effect of the colloidal interaction potential on the collector efficiency was examined. In addition, Scholl et al. [11] performed Brownian dynamics simulations to numerically simulate TIRM (total internal reflection microscopy) experiments for the motion of a sphere in a viscous fluid near to a wall. Numerous experimental techniques have been developed to study particle deposition [1,2]. The parallel-flow channel and the impinging jet cell are two widely used techniques for particle deposition experiments because of their numerous advantages, such as well possessing controlled and characterized hydrodynamic flow, allowing direct ob-

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servation of the deposition process by videomicroscopes, etc. The impinging jet cell technique pioneered by Dabros and van de Ven [12,13] was used for characterizing polymer adsorption [14], particle detachment [15], deposition of oil emulsions [16], and attachment of micron-sized bubbles [17]. Such a technique has been improved by Adamczyk and co-workers [18–20]. Experiments on particle deposition in parallel-plate channel flow were performed by Fattah et al. [21], Lüthi and Riˇcka [22], and Lüthi et al. [23] using latex particles and glass collector surfaces. These experimental works [21–23] focused on the spatial and temporal distributions of the deposited colloids that were obtained by optical microscopic imaging followed by detailed image processing and data extraction routines. In general, techniques such as RSA and ballistic deposition methods, which are based on the probability of particle deposition on a surface, are adequate for the analysis of irreversible particle deposition in the case of negligible flow and low surface coverage. The blocking process is driven by both the hydrodynamic and electrostatic (interparticle repulsion) effects when strong hydrodynamic interactions are present within the system. Hence, in the case of flowing dispersions (such as the case studied in the present work), Brownian dynamic simulation seems more suitable to incorporate the effect of hydrodynamic interactions on surface blocking. In the present work, an attempt has been made to study the deposition of colloidal particles from a pressure-driven flow in microchannels; it may be of interest for the surface fouling of microfluidic devices. A mathematical model based on the Brownian dynamics simulation technique is presented to compute the particle trajectories and, consequently, the surface coverage. The so-called “Blocking effect” that stems from the interactions between deposited and flowing particles is taken into consideration in the model development. To validate the model, experiments are carried out using the parallel-plate flow technique. In addition, the effects of the electrolyte concentration, flow velocity, and particle size on the particle surface coverage are studied.

2. Theory 2.1. Langevin equation The colloidal particle trajectory is governed by the Langevin equation, described by Ermack and McCammon [24] for flowing colloidal dispersions, dri =

Dij (t) · Fi (t) t + ∇rj · Dij (t)t kT + U(ri )t + (r)B ,

(1)

where dri is the position vector of the ith particle, Dij is the second-order mutual diffusivity tensor (due to the presence of the neighboring particles), U(ri ) is the particle velocity, Fi is the total force acting on the particle, (r)B is the random Brownian displacement of the particle, t is the time

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(a)

(b) Fig. 1. (a) Parallel-plate microchannel and associated coordinate system. (b) Forces acting on a moving particle i in the neighborhood of another particle j and the channel wall.

step (selected from the Brownian relaxation time of the particle), and kT is the thermal energy. 2.2. Hydrodynamic interactions The parallel plate channel geometry and the associated coordinate system are shown in Fig. 1a. Correction functions, named universal hydrodynamic correction functions (UHCC), are employed to relate the particle velocity to the fluid velocity. In this study, a 6-mm-wide and 0.3-mm-deep microchannel is used in the experiment, and hereby the width to depth ratio of the microchannel is 20; this allows us to assume a fully developed flow between two parallel plates. Thus, the corrected particle velocity in a parallel plate microchannel flow can be expressed as
 y y 3 2− , U = F (H, Hj ) Vm (2) 2 B B where Vm is the average flow velocity and B is the halfchannel depth. F (H, Hj ) is expressed as F (H, Hj ) = f3 (H )f (Hj 1 )f (Hj 2 ) · · · f (Hj N ),

(3)

where N is the number of neighboring particles. In the above expression, H is defined as the dimensionless particle–wall separation and Hj is defined as the dimensionless interparticle separation, and they are expressed as H=

y − ap , ap

Hj =

Rj − 2ap , ap

(4)

where Rj is the center-to-center separation distance between the particles and ap is the particle radius. Rj can be calculated from the computed particle positions. In Eq. (3), f3 (H ) is the universal hydrodynamic correction function for the motion a particle parallel to the wall, and f (Hj 1 ) to f (Hj N ) represent the hydrodynamic correction functions for two-particle interactions between the moving particle and its neighbors (1, 2, 3, . . . , N ). Analytical expressions for f3 (H ) and f (Hj ) are presented by Elimelech et al. [1]. Expressions for f (Hj ) are presented for the case of particle motion parallel and perpendicular to the line of centers connecting the particles, and are resolved onto the axis parallel to the channel wall. In dense suspensions, the diffusivity of a moving particle is affected by the neighboring particles. The concentration of particles near to the channel wall is of significance in particle deposition processes. In the simulation study, the minimum value of particle volume fraction φ (ratio of the volume of particles to the volume of simulation cell) of 0.5-µm particles based on the interparticle separation in the simulation box was found to be 0.05 after the equilibrium step (described in Section 3) that ensures steady values of potential energy of the system. Watzlawek and Nagele [29] have provided an algebraic relation for the ratio of the particle diffusivity to the self-diffusivity (estimated from the Stokes–Einstein equation) as a function of particle volume ratio. Using this relation, it can be calculated that the diffusivity ratio for φ = 0.05 is nearly 0.95. This indicates that the particle diffusivity deviates from the self-diffusivity at this volume fraction, and the modified diffusivity of a particle due to the presence of neighboring particles (also named mutual diffusivity) can be expressed as [1] Dij = Di I + kT

N 

Mij ,

(5)

j =1

where Di is the self-diffusivity of the ith particle, defined by the expression Di = kT /6πµap , I is the second-order unit tensor, and Mij is the second-order hydrodynamic mobility tensor that accounts for the interparticle hydrodynamic interactions. The complex expression for the hydrodynamic mobility tensor for the motion of two spheres in the vicinity of a wall is provided by Elimelech et al. [1]. N is the number of nearest neighbors of the ith particle. Hence, Eq. (5) is based on the superposition of hydrodynamic interactions of the neighboring particles on the ith particle. 2.3. External forces The colloidal transport to the channel wall is affected by various surface and body forces. In the present work, colloidal forces, which are constituted of by the van der Waals and the electric double layer (EDL) forces, and external forces such as gravity are taken into consideration. For particle deposition, the EDL forces can be either positive or negative, depending on the nature of the charges of the particle and the wall, while the van der Waals force is always

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attractive. The schematic of various forces acting on a moving particle in the vicinity of another particle and the channel wall is shown in Fig. 1b. The total force on the ith particle, Fi , is composed of Fi = FVDW + FEDL + FG ,

(6)

where FVDW , FEDL , and FG represent the van der Waals, EDL, and gravity forces, respectively. The colloidal forces are constituted of both the particle– wall and interparticle forces. In the present study, interparticle DLVO forces are expressed using the superposition rule [1]. This means that the effect of each nearest neighbor particle on the ith particle is superimposed on the particle– wall DLVO force to obtain the total DLVO force. The van der Waals force on the ith particle is given by [6] FVDW = −

Aλap (λ + 22.22h) eˆ y 6h2 (λ + 11.11h)2

+

N 

−

j =1

Aj λap (λ + 22.22hj )

eˆ ij , 12h2j (λ + 11.11hj )2

(7)

where eˆ y is the unit vector perpendicular to the channel wall, eˆ ij is the unit vector in the direction of the mutual separation between particles, and h and hj are the particle–wall separation and the interparticle separation, respectively, A and Aj are the Hamaker constants for the particle–wall interaction and the mutual interaction between particles, respectively, and λ is the London retardation wavelength. In the simulation, N is determined by providing a cutoff radius equal to 1.5 times the particle radius within which the colloidal interactions would be significant [8]. Similarly, the EDL force on the ith particle is given by [6]
2  kT 2γp γw exp(−κh)ˆey FEDL = 32πεκap e  N  + γp2 exp(−κhj )ˆeij , (8) j =1

with expressions for  

zeζp zeζw γp = tanh and γw = tanh , 4kT 4kT where ε is the permittivity of the medium, e is the fundamental electronic charge, and ζp and ζw are the particle and wall surface (zeta) potentials, respectively. In this study, both the particle and channel wall zeta potentials are assumed constant during particle deposition processes. κ is the Debye parameter, which is (inverse of EDL thickness), expressed as  2z2 e2 n∞ , κ= (9) εkT where z is the valency of the symmetric electrolyte (z+ = z− = z), n∞ is the bulk ionic concentration, and T is the absolute temperature.

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The gravity force is expressed as 4 FG = − πap3 (ρp − ρf )gˆey , (10) 3 where ρp and ρf are the particle and fluid densities, respectively. 2.4. Brownian motion The random Brownian displacement of a particle (due to collisions with the fluid molecules) is represented by a Gaussian distribution with zero mean and variance depending on the mutual diffusivity of the particles,  B B r r = 2Dij t, (11) where Dij is the mutual diffusivity of the ith particle [24] and t is the simulation time step, determined from the Brownian relaxation time of the particle, τBr [8], τBr =

ap2 6Di

.

(12)

The time step for simulation should be chosen to be sufficiently smaller than the Brownian relaxation time of the particle [8]. In the present study, a time step of 3 × 10−5 τBr was selected and the total number of time steps corresponding to the total simulation time (t) were 2 × 108 and 2.5 × 107 , respectively, for 0.25 and 0.5-µm particles.

3. Simulation The particle transport was simulated in a cubic simulation cell containing an arbitrary number of particles, 27, 64, 125, 216, etc. Computational tests were performed to determine the optimum number of particles (Np ) above which the variation in surface coverage is negligible. Periodic boundary conditions are applied on all the faces of the cell to eliminate the surface effects associated with the finite size of the cell [6]. This means that the simulation cell is replicated as an infinite lattice and as the particle leaves through one face of the cell, an image of the particle enters through the opposite face. The particle positions were computed according to Eq. (1) using a modified Euler approach; i.e., the predicted (superscript p) particle position is calculated from Eq. (1) from the current position and the parameters of the Langevin equation are determined at this position. The mean of the parameters at the current and the predicted positions is then used to compute the corrected (superscript c) particle position from the predicted positions. The position of a particle i at time (t + t) is computed based on the position of (N − 1) particles at time t. Once steady values over time for the potential energy of the system are achieved, the resulting particle configurations are selected as the starting configurations for the simulation (equilibrium step). The particle is assumed to be adsorbed on the surface once the primary energy minimum separation, H0 , is reached. H0 is the closest

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separation distance (dimensionless) between the particle and the wall. In the simulation study, computational tests were performed to estimate the influence of the choice of H0 on the surface coverage. It was found that the results are insensitive to the choice of H0 as long as H0
0.002. This means that a change in the value of H0 by 5 × 10−4 causes a change in the calculated surface coverage by 0.1%. Hence, H0 = 0.002 was used for all the simulation runs. The details of this algorithm are outlined in the following equations: p

p

rt+t = rt + (dr)t ,  Dij (rt ) · Fj (rt ) p t + ∇rj · Dij (rt )t (dr)t = kT

(13)

j

+ U(rt )t + (r)B ,

(14)

=  [Dij (rt ) + Dij (rt+t )p ] · [Fj (rt ) + Fj (rt+t )p ] t 4kT

(dr)ct j

1 ∇rj · Dij (rt ) + ∇rj · Dij (rt+t )p t 2 [U(rt ) + U(rt+t )p ] t + (r)B , + 2 rct+t = rt + (dr)ct . +

(15)

(a)

(16)

Based on the simulation results, the particle surface coverage is calculated using the expression θ=

πap2 Nd

(17) , S where Nd is the number of deposited particles, and S is the surface area of the simulation cell.

4. Experiment 4.1. Flow cell To validate the Brownian dynamics simulation results, particle deposition experiments were carried out using the parallel-plate flow technique. The flow cell consists of a glass microchannel in connection with two Teflon blocks. The parallel plate glass channel has length 100 mm, width 6 mm, and depth 0.3 mm, and it permits the colloids to be visualized at the channel inner surface and the bulk suspension with equal contrast. Owing to the small effective volume (180 mm3 ) of the cell, small amounts of suspension are required to perform the experiment for an extended period of time under various flowing conditions. 4.2. Experimental system A schematic of the experimental setup is shown in Fig. 2a. A Leica DMLM microscope with dark field illumination is used to visualize and image the colloidal particles

(b) Fig. 2. (a) Schematic of experimental setup. (b) Videomicroscopic image of 0.25-µm polystyrene latex particles captured using 50× objective under the experimental conditions C = 0.1 M and Re = 30.

deposited on the walls of the glass microchannel. Monodisperse suspensions of polystyrene latex particles 0.5 and 1 µm in diameter (Duke Scientific, USA) in NaCl electrolyte solutions were used in the experiment. The pH of the suspension was assumed to remain constant during the period of experiment. In the present study, a particle concentration of 2 × 1014 /m3 was employed in the experiment. Prior to the experiment, the glass microchannels were cleaned by soaking in acetone, sodium hydroxide, and hydrochloric acid for 20 min each followed by rinsing many times with deionized

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Table 1 Particle–wall and interparticle zeta potentials corresponding to various electrolyte (NaCl) concentrations NaCl concentration 0.1 M 0.01 M 0.001 M

Zeta potentials ζw

ζp

−18 mV −23 mV −28 mV

−14 mV −20 mV −24 mV

water. Particle suspensions were homogenized ultrasonically prior to the experiment. A peristaltic pump (Masterflex, USA) was used to pump the suspension continuously. The flow disturbance was reduced by a flow bunker and the flow rate was continuously monitored by a flow meter. The particle images (shown in Fig. 2b) delivered by the objective lens were captured by a digital camera (Leica DC300) and transferred to a PC for further image analysis. Initially, particle images were taken during an interval of 1 min, whereas at the later stages, images at 5-min intervals were captured. Particle images were processed and counted for the deposited particle number (Nc ) using Imagepro software. The particle surface coverage based on experimental data is determined as θ=

Fig. 3. Surface coverages versus time computed using different particle numbers. Other parameters used in simulation are ap = 0.25 µm, Re = 30, C = 0.01 M, ζw = −23 mV, ζp = −20 mV, A = 0.91 × 10−20 J, Aj = 0.38 × 10−20 J, and ρ = 1050 kg/m3 .

πap2 Nc

, (18) A where Nc is the deposited particle number and A is the view area of particle deposition experiment.

5. Results and discussion In this section, the effects of particle size, electrolyte concentration, and flow velocity (Reynolds number) on the particle surface coverage are discussed. The physical and electrochemical properties of NaCl electrolyte solution at room temperature (T = 298 K) were used for the simulation. The parameters for calculating the van der Waals and EDL forces, such as the Hamaker constant and zeta potential, were selected from [25–27]. In the present study, the Hamaker constants for the particle–wall and interparticle van der Waals interactions were chosen as 0.91 × 10−20 J (latex–NaCl–glass) and 0.38 × 10−20 J (latex–NaCl–latex), respectively. Particle and wall zeta potentials corresponding to the electrolyte concentrations considered in this study are shown in Table 1. As the pH of the suspension was assumed to remain constant during the period of the experiment, the particle and channel wall zeta potentials were also assumed to remain unchanged before and after the experiment. 5.1. Particle number independence test Fig. 3 shows the surface coverages computed with different particle numbers (Np ) for Reynolds number Re = 30, electrolyte concentration C = 0.01 M, and particle radius ap = 0.25 µm as a function of time. The computation was

Fig. 4. Deposition kinetics of 0.25-µm particles at different electrolyte concentrations. Other parameters used in simulation are Re = 30, A = 0.91 × 10−20 J, Aj = 0.38 × 10−20 J, and ρ = 1050 kg/m3 .

performed to ensure that the simulation results were independent of the choice of particle number. Simulations were carried out using Np = 27, 64, 125, and 216. The results corresponding to Np = 27 show a noticeable difference from the results of the other three particle numbers. It was found that the particle number Np  125 could produce particlenumber-independent results and hence Np = 125 was used for all simulation runs. 5.2. Effect of electrolyte concentration The effect of electrolyte concentration on particle deposition is presented in Fig. 4. Surface coverages of 0.25-µm particles at C = 0.1 and 0.001 M and Re = 30 are plotted. The solid lines represent the simulations results whereas the symbols denote the experimental data. As is evident from the figure, the surface coverage is larger for higher electrolyte

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Fig. 5. Deposition kinetics at different electrolyte concentrations as a function of flow Reynolds number. Other parameters used in simulation are τ = 1, ap = 0.25 µm, A = 0.91 × 10−20 J, Aj = 0.38 × 10−20 J, and ρ = 1050 kg/m3 .

Fig. 6. Deposition kinetics of 0.25 µm particles at different Reynolds numbers. Other parameters used in simulation are C = 0.1 M, ζw = −18 mV, ζp = −14 mV, A = 0.91 × 10−20 J, Aj = 0.38 × 10−20 J, and ρ = 1050 kg/m3 .

concentrations. This can be attributed to the repulsive double layer interactions between the same charge polarity (negative) of the latex particles and glass wall. A linear adsorption regime is observed at the initial stages, after which, a tendency for saturation is noted. Similar kinetic saturations of the interfaces were observed by Adamczyk [6] and Yang et al. [28] in deposition experiments using the impinging jet and parallel-plate channel flow techniques. The reduction in particle deposition at the later stages can be explained by taking into account the surface blocking, which is dependent on the combined effect of hydrodynamic (particle velocity) and EDL interactions. The former cause a reduction in particle velocity in the vicinity of the neighboring particles, with the slower particle transported to larger separation distances. The double-layer interactions are dependent on electrolyte concentration and are reduced at higher electrolyte concentrations. Fig. 5 shows the variation of surface coverages with the Reynolds number corresponding to τ = 1 for three different electrolyte concentrations. For the range of concentrations studied, the deposition rate decreases significantly with an increase in the Reynolds number. The effect of the Reynolds number is more pronounced at C = 0.1 M than at the two lower concentrations. This is due to the stronger Blocking effect resulted from more deposited particles. Further, the simulation results are found to be in quantitative agreement with the experimental results.

and υ is the kinematic viscosity of the fluid. It can be observed that an increase in the flow intensity decreases the deposition rate significantly. One can observe from Fig. 6 that the slope of the curve for Re = 60 is much less than that for Re = 30, particularly at the late stages of deposition. This shows that when τ > 0.5, the flow conditions strongly influence deposition, leading to saturation of the surface. This indicates that a stronger blocking effect is present in the latter case due to the presence of increasing importance of hydrodynamic interactions. Hence, most of the particles are transported without being able to contact the wall. Adamczyk et al. [6], however, have observed an increase in surface coverage at higher Reynolds numbers in impinging jet flows. This can be attributed to the fact that flow velocity facilitates the deposition in the impinging jet cell flows; whereas in the parallel plate channel geometry, the flow swipes particles away from the channel wall.

5.3. Effect of hydrodynamic flow The effect of the Reynolds number on the surface coverage is presented in Fig. 6. Surface coverages of 0.25-µm particles corresponding to Re = 30 and 60 at C = 0.1 M are plotted. The flow Reynolds number is defined as Re = 2V (2B)/υ, where V is the average fluid velocity; 2B is the depth of the microchannel (in the case of parallel plate channels, the hydraulic diameter Dh = 2 × depth of the channel),

5.4. Effect of particle size The effect of particle size on deposition rate is illustrated in Figs. 7 and 8. Surface coverages at C = 0.1 M and Re = 30 for two different particle sizes (ap = 0.25 and 0.5 µm) are presented in Fig. 7. A significant increase in surface coverage for smaller particles can be observed. This can be explained in light of the fact that the colloidal interactions, which are surface forces, become increasingly important with reduction in particle size. Hence, the effect of the surface forces on the particle transport to the surface is stronger for smaller particles. Furthermore, since the area occupied by the deposited particles is expressed as πap2 Nd , being proportional to the square of particle radius, the available surface area for particle deposition is reduced for large particles. As a result, further deposition of 0.5 µm particles is prevented due to stronger blocking effects. Hence, there is no significant increase in deposition for 0.5-µm particles

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Fig. 7. Deposition kinetics for different particle sizes. Other parameters used in simulation are Re = 30, C = 0.1 M, ζw = −18 mV, ζp = −14 mV, A = 0.91 × 10−20 J, Aj = 0.38 × 10−20 J, and ρ = 1050 kg/m3 .

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Fig. 9. Dimensionless particle–wall interaction energy profiles corresponding to various electrolyte concentrations. ap = 0.25 µm, A = 0.91 × 10−20 J, Aj = 0.38 × 10−20 J.

5.5. Comparison of theoretical predictions with experimental results

Fig. 8. Effect of particle size on particle deposition kinetics. Other parameters used in simulation are τ = 1, C = 0.1 M, ζw = −18 mV, ζp = −14 mV, A = 0.91 × 10−20 J, Aj = 0.38 × 10−20 J, and ρ = 1050 kg/m3 .

at the late stages of deposition compared to 0.25-µm particles. The simulation results are in good agreement with the experimental data, indicating that the two-body superposition approach employed in the computation of interparticle forces is a good approximation when dealing with dilute suspensions. Fig. 8 shows the variation of surface coverage with the Reynolds number corresponding to two different particle sizes for C = 0.1 M and τ = 1. At higher flow intensities, the convection effects circumvent the effect due to colloidal forces and the particles are transported in the microchannel without being deposited. The difference in surface coverage for particle sizes 0.25 and 0.5 µm decreases with an increase in the Reynolds number. This is indicated by the higher slope of the curve corresponding to ap = 0.25 µm.

In the present study, good agreement exists between theory and experiment for the range of electrolyte concentrations and flow Reynolds numbers studied. There are two cases in particle deposition studies that should be mentioned in this regard; (i) favorable deposition, where the combined van der Waals and EDL interactions between the particle and wall are attractive, and (ii) unfavorable deposition where a repulsive energy barrier exists in the interaction energy profile. In the literature, good agreement between theory and experiment has been reported when deposition occurred under favorable conditions (Adamczyk et al. [6], Elimelech et al. [30]). In contrast, large discrepancies are usually found between theoretical and experimental deposition rates under unfavorable deposition conditions (Sjollema and Busscher [31], Elimelech and O’Melia [32]). The reasons for these discrepancies are attributed to various stochastic effects such as particle and wall surface roughness, surface charge heterogeneity and nonuniformity in the particle and wall surface (zeta) potential, etc. Fig. 9 shows the DLVO estimations based on the surface parameters selected in this study, where the dimensionless interaction energy (φ/kT ) is plotted against the dimensionless particle–wall separation distance. It can be noted that for C = 0.1 and 0.01 M, the resultant particle–wall interaction energy is attractive due to weak repulsive double-layer interactions. Hence the agreement between theory and experiment for the two lower concentrations is not surprising given that the deposition occurred under such favorable conditions. When C = 0.001 M, a repulsive energy barrier of 6.36kT is present, and thus the deposition can be considered under an unfavorable condition. However, as the energy barrier is not high, any mechanisms such as Brownian motion that have not been included in conventional models

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may be responsible for the occurrence of particle deposition. From an experimental viewpoint, the disturbances due to hydrodynamic flow and interaction, surface interaction forces, and Brownian effects would possibly cause particles to overcome such a small energy barrier and thus to achieve deposition. In addition, it should be pointed out here that the surface properties of the particles and wall (zeta potentials and Hamaker constant) were not directly measured in the experiment and instead were selected from the literature. The differences between the selected and the true values of the surface properties can cause variations in the height of the energy barrier, resulting in particle deposition.

6. Conclusions The irreversible adsorption of colloidal particles from the pressure-driven flow in a parallel-plate microchannel has been investigated in this paper. The Brownian dynamics simulation technique based on the stochastic Langevin equation was employed for computation of particle surface coverage. Deposition experiments using the parallel-flow technique were carried out to compare the simulation results, and good agreement was found for the range of Reynolds numbers studied. Further, the electrolyte concentration was found to influence the surface coverage significantly at lower Reynolds numbers; however, this effect is reduced at higher Reynolds numbers due to the increased particle velocity and scattering of the moving particles in the vicinity of the deposited particles. In addition, as the particle size is reduced, the particle deposition increases due to increasing contributions of the colloidal interaction forces. Again, the effect of particle size starts decreasing at high Reynolds numbers.

Acknowledgment Harikrishnan N. Unni gratefully acknowledges a Ph.D. scholarship from Nanyang Technological University.

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